We show that the algorithm converges globally to an \(\epsilon \) -approximate stationary point of (REF ) in no more than \(O(\epsilon ^{-4})\)  iterations and locally converges to an approximate local minimizer at a sublinear rate under mild assumptions.
The proof uses an easy-to-compute Lyapunov function that does not require a (typically unknown) local minimizer.
We supplement the analysis by showing (empirically) that convergence fails if the proximal weights in the algorithm are not chosen appropriately.
The algorithm can be interpreted as a nonconvex constrained extension of the Jacobi algorithms proposed in [1]}, [2]}, [3]}, [4]}, and as a Jacobi extension of the Gauss-Seidel algorithm for nonconvex problems proposed in [5]}.
In contrast to the majority of existing approaches for nonconvex problems,
we provide a practical and automatic parameter tuning scheme, an open-source Julia implementation (that can be downloaded from https://github.com/exanauts/ProxAL.jl), and an empirical demonstration of convergence on a large-scale multi-period optimal power flow test case.
